I have had the pleasure of reviewing for MAA Reviews a number of books connected to the Hamiltion Ricci flow and its role in Perelman’s solution of the 3-dimensional Poincaré conjecture: surely a story for the ages, not just because of the scope of Perelman’s achievement, but also because of the controversy surrounding his work when it first appeared on the radar screen and later because of Perelman’s now famous refusal of any and all accolades the mathematical establishment sought to bestow on him, including the Fields Medal and the Clay Prize.

Regarding the early controversy it was the case that, when the mathematical jungle drums first started up, a circle of Chinese mathematicians with Yau at its apparent center placed themselves at odds with Perelman in connection with certain priority matters. These acrimonies were made extra-sticky by Perelman’s work having been launched entirely on the internet, in the form of three rather idiosyncratic articles, often containing sketches rather than full proofs. This ugly business was in due course settled in favor of Perelman (obviously), and his arguments were found to be entirely on the mark. In any event what would otherwise be the story of a heroic quest vouchsafed only to mathematicians and perhaps a tiny number of physicists is now proper fodder for the *New Yorker*. Given Perelman’s insistence on isolation and anonymity there is no small amount of irony in all this.

In any event, in connection with Ricci flow, I have reviewed in this column the books, *Ricci Flow and the Poincaré Conjecture*, by John Morgan and Gang Tian, and *Hamilton’s Ricci Flow*, by Bennett Chow, Peng Lu, and Lei Ni. And now it is my pleasant task to review *Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture*, by Qi S. Zhang of U. C. Riverside, and I want to start by taking note of a number of synchronicities I find particularly appealing: Zhang was introduced to the subject of Ricci flow by Bennett Chow and Lei Ni in a 2005 workshop on geometric analysis, while it was Gang Tian who recommended the Chinese version of the 2008–2009 Peking / Nanjing lecture notes that evolved into part of the book under review for publication by Science Press Beijing. Additionally, two chapters (the second and the fourth) of *Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture *spring from a UCR graduate course one of whose note-takers, Jennifer Burke-Loftus, was my student in a number of courses when an undergraduate at my university. It is indeed a small world.

So, Chapters 2 and 4, specifically focused on Sobolev inequalities, are *eo ipso* introductory affairs with respect to the main thrust of the book: these two chapters should be entirely within the province of sufficiently advanced analysis or PDE graduate students. (I always marvel at young scholars like Jennifer who opt to go to the mat with hard analysis, PDE, and so forth: what fortitude!)

Chapter 1 is a well-written and informative introduction; here is a sample: “… a Sobolev inequality states [roughly] that if the derivative of a function is integrable in a certain sense (L^{p}, etc.), then the function itself has better integrability. It lies in the foundation of modern analysis… On the other hand a Sobolev inequality will also yield interesting partial differential equations *via* minimizing the Sobolev constants. It can also reveal useful information on the underlying space or manifold. This last property is the focus of this book.”

Thus, with the game defined early on in this way, Zhang sandwiches in an important (if somewhat austere: have another source handy, maybe) Chapter 3, innocuously titled, “Basics of Riemannian Geometry.” Then, with Chapter 5, gets airborne. To wit, Ricci flow is discussed in the fifth chapter, quickly followed in Chapter 6 by unabashedly *avant garde* stuff: “Perelman’s entropies and Sobolev inequality.” I want to single out §6.4 as particularly piquant: “Harnack’s inequality for the conjugate heat kernel.” This by itself indicates the potential breadth of this material (its depth already being obvious).

Subsequently Chapter 7, “Ancient κ-solutions and singularity analysis,” continues to develop the all-important heat kernel theme, and obviously begins to get down to some of the geometric nitty-gritty. By the way, here’s something relevant from p. 180 of the aforementioned book by Morgan and Tian: “An ‘ancient solution’ is a Ricci flow (M, g(t)) defined for –κ < t ≤ 0 such that for each t, (M, g(t)) is a connected, complete, non-flat Riemannian manifold whose curvature is bounded and non-negative.” Beautiful stuff, no?

And then Zhang’s seventh chapter culminates in § 7.5, “Singularity analysis of 3-dimensional Ricci flow,” introduced as follows: “The main result of this section is the next theorem which says that a space time cube in a 3 dimensional Ricci flow resembles a κ solution provided that the scalar curvature at one point of the cube is sufficiently large.” Very tantalizing. (To find out more about the mysterious κ see p. 209 of the book under review.)

Finally Chapter 8 discusses — briefly — “Sobolev inequality with surgeries” and Chapter 9, the book’s last chapter, is titled, yes, “Applications to the Poincaré conjecture.”

It clear as vodka that, as Zhang advertises in the Preface, “[t]he first half of the book is aimed at graduate students and the second half is intended for researchers.” With some good timing the same reader can start as one and continue as the other. *Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture* looks to be a very important contribution to the genre.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.